The smallest common extension of a sequence of models of ZFC

Lev Bukovský; Jaroslav Skřivánek

Commentationes Mathematicae Universitatis Carolinae (1994)

  • Volume: 35, Issue: 4, page 745-752
  • ISSN: 0010-2628

Abstract

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In this note, we show that the model obtained by finite support iteration of a sequence of generic extensions of models of ZFC of length ω is sometimes the smallest common extension of this sequence and very often it is not.

How to cite

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Bukovský, Lev, and Skřivánek, Jaroslav. "The smallest common extension of a sequence of models of ZFC." Commentationes Mathematicae Universitatis Carolinae 35.4 (1994): 745-752. <http://eudml.org/doc/247633>.

@article{Bukovský1994,
abstract = {In this note, we show that the model obtained by finite support iteration of a sequence of generic extensions of models of ZFC of length $\omega $ is sometimes the smallest common extension of this sequence and very often it is not.},
author = {Bukovský, Lev, Skřivánek, Jaroslav},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {model of ZFC; generic extension; rigid Boolean algebra; hereditary $M$-definable; generic extension; iterated forcing; rigid Boolean algebra; finite support iteration; Cohen real},
language = {eng},
number = {4},
pages = {745-752},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {The smallest common extension of a sequence of models of ZFC},
url = {http://eudml.org/doc/247633},
volume = {35},
year = {1994},
}

TY - JOUR
AU - Bukovský, Lev
AU - Skřivánek, Jaroslav
TI - The smallest common extension of a sequence of models of ZFC
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1994
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 35
IS - 4
SP - 745
EP - 752
AB - In this note, we show that the model obtained by finite support iteration of a sequence of generic extensions of models of ZFC of length $\omega $ is sometimes the smallest common extension of this sequence and very often it is not.
LA - eng
KW - model of ZFC; generic extension; rigid Boolean algebra; hereditary $M$-definable; generic extension; iterated forcing; rigid Boolean algebra; finite support iteration; Cohen real
UR - http://eudml.org/doc/247633
ER -

References

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  1. Blass A., The model of set generated by countable many generic reals, J. Symbolic Logic 46 (1981), 732-752. (1981) MR0641487
  2. Bukovský L., Characterization of generic extensions of models of set theory, Fund. Math. 83 (1973), 35-46. (1973) MR0332477
  3. Bukovský L., A general setting of models extensions, International Workshop on Set theory, Abstracts of the talks, Marseilles-Luminy (1990). (1990) 
  4. Ciesielski K., Guzicki W., Generic families and models of set theory with the axiom of choice, Proc. Amer. Math. Soc. 106 (1989), 199-206. (1989) Zbl0673.03042MR0994389
  5. Jech T., Set Theory, Academic Press New York (1978). (1978) Zbl0419.03028MR0506523
  6. Solovay R., Tennenbaum S., Iterated Cohen Extensions and Souslin`s Problem, Ann. of Math. 94 (1971), 201-245. (1971) Zbl0244.02023MR0294139
  7. Štěpánek P., Embeddings and automorphisms, Handbook of Boolean algebras J.D. Monk and R. Bonnet North-Holland Amsterdam (1989), 607-635. (1989) MR0991604
  8. Vopěnka P., Hájek P., The Theory of Semisets, Academia Prague (1972). (1972) MR0444473

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